If you are going to do your own reloading I would take the 22-250. It is a more flexible round.
If you want .223 level of recoil for a given application you can load it light. If you want more zip you can load it hot.
.223 ammo is cheaper, but 22-250 isn't that hard to find, especially if you are going to reload the ammo price difference is even less. Yes the 22-250 will use more powder, BUT seriously, with relatively low volume shooting (any hunting rifled compared to skeet/trap shooting), the amount of powder that you go through is minimal. Is there a difference?? Yes. Does it matter, likely not.
Either way you are not going to go wrong, each has a niche. If you are a reloader or plan to reload, 22-250 hands down, if you are shooting factory ammo think about the actual amount of shooting that you are going to do in a year, and look at the difference in ammo prices. The .223 might be the best choice for you.